Uniwersytet Jagiello´ nski
Wydzia l Matematyki i Informatyki Instytut Informatyki
Anna Kulig
Nonlinear Evolution Inclusions and Hemivariational Inequalities
for Nonsmooth Problems in Contact Mechanics
PhD Thesis
written under the supervision of prof. dr hab. Stanis law Mig´orski
Krak´ ow 2009
”La Meccanica `e il paradiso delle scienze matematiche, perch´e con quella si viene al frutto matematico.”
– Milano, 1483.
p ”Mechanics is the paradise of the mathematical sciences, because by means of it one comes to the fruits of mathematics.” y Leonardo da Vinci (1452–1519)
Abstract. The dissertation deals with second order nonlinear evolution inclusions, hyperbolic hemivariational inequalities and their applications. First, we study a class of the evolution inclusions involving a Volterra integral operator and considered within the framework of an evolution triple of spaces. Combining a surjectivity result for mul- tivalued pseudomonotone operators and the Banach Contraction Principle, we deliver a result on the unique solvability of the Cauchy problem for the inclusion. We also provide a theorem on the continuous dependence of the solution to the inclusion with respect to the operators involved in the problem. Next, we consider a class of hyper- bolic hemivariational inequalities and embed these problems into a class of evolution inclusions with the multivalued term generated by the generalized Clarke subdiffer- ential for nonconvex and nonsmooth superpotentials. Finally, we study a dynamic frictional contact problem of viscoelasticity with a general constitutive law with long memory, nonlinear viscosity and elasticity operators and the subdifferential boundary conditions. We deal with various aspects of the modeling of these contact problems and provide several examples of nonmonotone subdifferential boundary conditions which illustrate the applicability of our findings.
Keywords. Hemivariational inequality, contact, friction, nonmonotone, hyperbolic, viscoelasticity, dynamic, evolution inclusion, nonsmooth, nonconvex, multivalued, pseudomonotone operator, hemicontinuous, subdifferential, existence, uniqueness, mo- deling.
2010 Mathematics Subject Classification (MSC2010): 35A23, 35L70, 35L86, 35L90, 35R05, 35R70, 45P05, 47H04, 47H05, 47H30, 47J05, 47J20, 58E35, 70E18, 74D10, 74H20, 74H25, 74H30, 74L10, 74L15, 74M10, 74M15.
Contents
1. Introduction 5
2. Preliminaries
2.1 Lebesgue and Bochner-Sobolev spaces . . . 9
2.2 Single-valued and multivalued operators . . . 12
2.3 Clarke’s generalized subdifferential . . . 14
3. Second order nonlinear evolution inclusions 3.1 Problem statement . . . 23
3.2 Evolution inclusion of Problem Q . . . 27
3.3 Main result for nonlinear evolution inclusion . . . 38
4. A convergence result for evolution inclusions . . . 42
5. Evolution hemivariational inequalities 5.1 Function spaces for contact problems . . . 46
5.2 Physical setting of the problem . . . 47
5.3 Weak formulation of the problem . . . 50
5.4 Evolution inclusion for hemivariational inequality . . . 54
5.5 Unique solvability of hemivariational inequality . . . 71
6. Applications to viscoelastic mechanical problems 6.1 Examples of constitutive laws with long memory . . . 75
6.2 Examples of subdifferential boundary conditions . . . 76
7. Appendix . . . 94
8. References . . . 98
1 Introduction
An important number of problems arising in Mechanics, Physics and Engineering Science lead to mathematical models expressed in terms of nonlinear inclusions. For this reason the mathematical literature dedicated to this field is extensive and the progress made in the last decades is impressive. It concerns both results on the exis- tence, uniqueness, regularity and behavior of solutions for various classes of nonlinear inclusions as well as results on numerical approach of the solution for the correspond- ing problems.
The framework of evolution inclusion allows to describe dynamical systems with multivalued discontinuities and therefore this framework is more general than evolu- tion equations. However, the great advantage of this framework over other approaches is, that physical interaction laws, such as contact and friction in mechanics can be formulated as set-valued force laws and can be incorporated in the formulation. We will therefore use the framework of nonlinear evolution inclusions in this thesis to study existence properties of nonsmooth systems which naturally arise in mechanics with inequality constraints. The abstract problem under investigation is the following second order evolution inclusion
u′′(t) + A(t, u′(t)) + B(t, u(t)) + Z t
0
C(t− s)u(s) ds + + F (t, u(t), u′(t))∋ f(t) a.e. t ∈ (0, T ), u(0) = u0, u′(0) = u1,
(∗)
where A, B : (0, T )× V → V∗ are nonlinear operators, C(t) is a bounded linear operator from V to its dual, for t ∈ (0, T ), F : (0, T ) × V × V → 2Z∗ denotes a multivalued mapping, f ∈ L2(0, T ; V∗), u0 ∈ V , u1 ∈ H, V and Z are reflexive Banach spaces with V ⊂ Z compactly, H is a Hilbert space such that Z ⊂ H and 0 < T <∞.
More precisely, we focus on the existence and uniqueness results for the Cauchy problem (∗). The latter is defined in the framework of an evolution triple of spaces. We approach the problem by reducing its simplified version to the first order nonlinear evolution inclusion through the introduction of the integral operator and by applying a result on the surjectivity of multivalued operators. Later we use the Banach Con- traction Principle to a suitable operator and show a result on unique solvability of the evolution inclusion (∗). We remark that in order to prove uniqueness of solutions we need, on one hand, some restrictive hypotheses on the multivalued term, and on the other hand these hypotheses should be quite general to cope with the multifunctions which appear in the contact problems.
The Mathematical Theory of Contact Mechanics has made recently impressive progress due to the development in the field of Inequality Problems. In the lat- ter we can distinguish two main directions: variational inequalities connected with convex energy functions and hemivariational inequalities connected with nonconvex energies. The variational inequalities have a precise physical meaning and they ex- press the principles of virtual work and power introduced by Fourier in 1823. The
prototypes of boundary value problems leading to variational inequalities are the Signorini-Fichera problem and the friction problem of elasticity. For variational in- equalities the reader is referred to monographs of Duvat and Lions [27], Hlav´aˇcek et al. [36], Kikuchi and Oden [44], Kinderlehrer [45] and Panagiotopoulos [77], among others. The notion of hemivariational inequality is based on the generalized gradient of Clarke-Rockafellar [21] and has been introduced in the early 1980s by Panagiotopou- los [77, 78] to describe several important mechanical and engineering problems with nonmonotone phenomena in solid mechanics. Such inequalities appear in the mode- ling of the constitutive law and/or the boundary conditions. The nonsmooth and nonconvex nature of energy potentials and the resulting multivalued character of me- chanical laws challenge the extension of the existing results for smooth and convex potential systems to evolution inclusions with multifunctions which are of the Clarke subdifferential form. For convex potentials the hemivariational inequalities reduce to the variational inequalities.
The evolution hemivariational inequalities have been studied for parabolic prob- lems by Miettinen [55] who employed the regularization method with the Galerkin technique, by Carl [15, 14] (who adapted the Rauch method of [87]) and Papageor- giou [82] who both combined the method of lower and upper solutions with truncation and penalization techniques. Moreover, Liu [53] obtained existence result for parabolic hemivariational inequalities with an evolution operator of class (S)+ and Miettinen and Panagiotopoulos [56] and Mig´orski and Ochal [63] have treated the problem using a regularized approximating model. The existence and convergence results for first order evolution hemivariational inequalities can be found in Mig´orski [59].
The hyperbolic hemivariational inequalities arising in nonlinear boundary value problems have been studied by Panagiotopoulos [78, 79], Panagiotopoulos and Pop [80]
who used the Galerkin method as well as Gasi´nski [30] and Ochal [75] who employed a surjectivity result for multivalued operators. The existence results for second or- der nonlinear evolution inclusions can be found in Ahmed and Kerbal [2], Bian [12], Mig´orski [57, 58], Papageorgiou [81], and Papageorgiou and Yannakakis [83, 84], while the existence of solutions to the dynamic hemivariational inequalites of second order has been studied by Guo [33], Kulig [48], Liu and Li [51], Mig´orski [60, 61, 62], Mig´orski and Ochal [65], Park and Ha [85] and Xiao and Huang [100]. A general method for the study of dynamic viscoelastic contact problems involving subdiffer- ential boundary conditions was presented in Mig´orski and Ochal [66]. Within the framework of evolutionary hemivariational inequalities, this method represents a new approach which unifies several other methods used in the study of viscoelastic con- tact and allows to obtain new existence and uniqueness results. Recent books and monographs on mathematical theory of hemivariational inequalities include Carl and Motreanu [16], Goeleven et al. [31], Haslinger et al. [35], Mig´orski et al. [70], Motre- anu and Panagiotopoulos [71], Naniewicz and Panagiotopoulos [73], Panagiotopou- los [77, 78], and we refer the reader there for a wealth of additional information about these and related topics. The results on Mathematical Theory of Contact Mechanics can be found in several monographs, e.g. Eck et al. [28], Han and Sofonea [34], Shillor et al. [93], Sofonea et al. [95] and Sofonea and Matei [96].
In the thesis the hemivariational inequalities under investigation represent a par-
ticular case of nonlinear inclusions associated to the Clarke subdifferential operator.
Specyfing the spaces V , Z and H as suitable Sobolev and L2 spaces defined on an open bounded subset Ω of Rd, considering the potential contact surface ΓC as a measurable part of the boundary of Ω and introducing an appropriate multivalued mapping F , it can be seen that every solution to the evolution inclusion (∗) satisfies the hyperbolic hemivariational inequality of the form
hu′′(t) + A(t, u′(t)) + B(t, u(t)) + Z t
0
C(t− s)u(s) ds − f(t), vi + +
Z
ΓC
g0(x, t, γu(t), γu′(t), γu(t), γu′(t); γv, γv) dΓ≥ 0 for all v∈ V, a.e. t ∈ (0, T ), u(0) = u0, u′(0) = u1,
(∗∗)
where g0 denotes the generalized directional derivative of a (possibly) nonconvex function g in the sense of Clarke, γ is a trace map and h·, ·i stands for the duality pairing between V∗ and V . For the definitions of the function g and the multivalued mapping F which give a passage from (∗) to the hemivariational inequality (∗∗), we refer to Section 5.4.
The goals and the results of the thesis are following. First, we establish a result on unique solvability of the Cauchy problem for the second order evolution inclusion (∗). The inclusion (∗) without the Volterra memory term and with time independent operator B has been studied in Denkowski et al. [24] (with F : (0, T )× H × H → 2H), Mig´orski and Ochal [67] in a case B is linear, continuous, symmetric and coercive operator, and in Mig´orski [60], and Park and Ha [85] in a case B is linear, continuous, symmetric and nonnegative. Now, we treat the problem (∗) with a nonlinear Lips- chitz operator B(t,·), and with a linear and continuous kernel operator C(t) in the memory term. We underline that none of the results on nonlinear evolution inclusions in [2, 12, 57, 58, 81, 83, 84, 98] can be applied in our study because of their restrictive hypotheses on the multivalued term which was supposed to have values in H. For the hemivariational inequalities and the contact problems, the associated multival- ued mapping has values in the space dual to Z which is larger than H. Moreover, we have employed a method which is different than those of [24, 60, 67, 85] and which combines a surjectivity result for pseudomonotone operators with the Banach Con- traction Principle. We obtain results on local and global (under stronger hypotheses on the multifunction) unique solvability of the evolution inclusion (∗).
Next, we provide a result on the continuous dependence of the solution to (∗) with respect to the operator A, B and C. It is shown that the sequence of the unique solutions to (∗) corresponding to perturbed operators Aε, Bε and Cε converges in a suitable sense to the unique solution corresponding to unperturbed operators A, B and C. This result is of importance from the mechanical point of view, since for vanishing relaxation operator, it indicates that the nonlinear viscoelasticity for short memory materials may be considered as a limit case of nonlinear viscoelasticity with constitutive law with long memory. This convergence result holds for the whole spectrum of nonmonotone contact conditions which we describe in this work.
Subsequently, we consider the class of evolution hemivariational inequalities of second order of the form (∗∗). Our study includes the modeling of a mechanical prob- lem and its variational analysis. We derive the hemivariational inequality (∗∗) for the displacement field from nonconvex superpotentials through the generalized Clarke subdifferential. The mechanical properties are described by a general constitutive law which include the Kelvin-Voigt law and a viscoelastic constitutive law with long memory. The novelty of the model is to deal with nonlinear elasticity and viscosity operators and to consider the coupling between two kinds of nonmonotone possibly multivalued boundary conditions which depend on the normal (respectively, tangen- tial) components of both the displacement and velocity. The new results concern the existence, uniqueness and regularity of the weak solution to the hemivariational in- equality (∗∗) which are obtained by embedding the problem into a class of evolution inclusions of the form (∗) and by applying the results obtained for (∗). To the author’s best knowledge the results obtained for hemivariational inequalities seem to be new even for the case when all/some of the potentials involved in the boundary conditions are convex functions. We also remark that the question on uniqueness of solutions to a general form of hemivariational inequality (∗∗) remains open.
Finally, in order to illustrate the cross fertilization between rigorous mathematical description and Nonlinear Analysis on one hand, and modeling and applications on the other hand, we provide examples of constitutive laws with long memory as well as several examples of contact and friction subdifferential boundary conditions. We mention that our formulation of multivalued boundary conditions covers, as particular cases, the following conditions used recently in the literature: frictionless contact, the nonmonotone normal compliance condition, the simplified Coulomb friction law, the nonmonotone normal damped response condition, the viscous contact with Tresca’s friction law, the viscous contact with power-law friction boundary condition, the version of dry friction condition, the nonmonotone friction conditions depending on slip and slip rate, and the sawtooth laws generated by nonconvex superpotentials. We will also show how a suitable choice of the multivalued term in the evolution inclusion leads to different types of boundary conditions.
The thesis is organized as follows. In Section 2 we recall some preliminary material which is needed in the work. In Section 3 we study a class of second order nonlinear evolution inclusions involving a Volterra integral operator in the framework of evolu- tion triple of spaces. For this class we give a result on the existence and uniqueness of solutions to the Cauchy problem for the inclusion under investigation. Section 4 is devoted to the study of the dependence of the solution to the abstract nonlinear evolution inclusion on the operators involved in the problem. In Section 5 we establish the link between a nonlinear evolution inclusion and the hemivariational inequality (HVI), and we apply results of Section 3 to the viscoelastic contact problem with a memory term. The review of several examples of contact and friction subdifferential boundary conditions which illustrates the applicability of our results is provided in Section 6. Section 7 contains a few results from functional analysis that are often used in the text.
A portion of the thesis concerning a mathematical model which describes dynamic viscoelastic contact problems with nonmonotone normal compliance condition and the
slip displacement dependent friction has been by published by the author in [48].
2 Preliminaries
In this section we provide the background material which will be needed in the sequel.
We summarize some results from the theory of vector-valued function spaces, briefly recall notions for classes of operators of monotone type, and present basic facts from the theory of the Clarke generalized differentiation of locally Lipschitz functions.
2.1 Lebesgue-Bochner and Sobolev spaces
In this part we recall some results from the theory of vector-valued function spaces which will be used in the sequel. For the details we refer to basic monographs of Adams [1], Br´ezis [13], Denkowski et al. [23, 24], Droniou [25], Evans [29], Gris- vard [32], Hu and Papageorgiou [37], Lions [52], Showalter [94] and Zeidler [99].
Let X be a Banach space with a normk · kX, let X∗ be its dual, and leth·, ·iX∗×X
denote the duality pairing between X∗ and X. Let 0 < T < ∞ and 1 ≤ p ≤ ∞. We denote by Lp(0, T ; X) the space (equivalent classes) of measurable X-valued functions v : (0, T )→ X such that kv(·)k belongs to Lp(0, T ; R) with
kvkLp(0,T ;X) =
Z T
0 kv(t)kpXdt
1/p
if 1≤ p < ∞, ess sup
0≤t≤T ku(t)kX if p =∞.
(1)
The space C(0, T ; X) comprises of all continuous X-valued functions v : [0, T ] → X with
kvkC(0,T ;X)= max{ kv(t)kX | t ∈ [0, T ] }.
Basic properties of the Lebesgue space Lp(0, T ; X) of Banach space valued func- tions are formulated below.
Proposition 1 Let X and Y be Banach spaces. We have the following results (i) The space Lp(0, T ; X) is a Banach space with respect to the norm (1) for p ∈
[1,∞].
(ii) If X is a Hilbert space with scalar producth·, ·iX, then L2(0, T ; X) is also a Hilbert space equipped with the scalar product
hhu, viiL2(0,T ;X)= Z T
0 hu(t), v(t)iXdt.
(iii) If X is a reflexive, separable Banach space and p ∈ (1, ∞), then Lp(0, T ; X) is reflexive, separable and (Lp(0, T ; X))∗ ≃ Lq(0, T ; X∗), where 1p + 1q = 1, and L1(0, T ; X) is separable with (L1(0, T ; X))∗ ≃ L∞(0, T ; X∗).
(iv) Let 1≤ r ≤ p < ∞. If the embedding X ⊂ Y is continuous, then the embedding Lp(0, T ; X) ⊂ Lr(0, T ; Y ) is also continuous. For the embedding Lp(0, T ; X) ⊂ Lr(0, T ; X), we have
kvkLr(0,T ;X) ≤ Tp−rpr kvkLp(0,T ;X) for all v∈ Lp(0, T ; X).
(v) If 1 ≤ p ≤ ∞ and {vn, v} ⊂ Lp(0, T ; X), vn→ v in Lp(0, T ; X), then there exists a subsequence {vnk} ⊂ {vn} such that vnk(t)→ v(t) in X for a.e. t ∈ (0, T ) and kvnk(t)kX ≤ h(t) with h ∈ Lp(0, T ).
(vi) If 1≤ p ≤ ∞ and X is a reflexive, separable Banach space, then for any bounded sequence {vn} in Lp(0, T ; X), there exists v ∈ Lp(0, T ; X) and a subsequence {vnk} ⊂ {vn} weakly convergent in Lp(0, T ; X) to v, i.e.
Z T
0 hvnk(t), w(t)iX∗×Xdt→ Z T
0 hv(t), w(t)iX∗×Xdt for all w∈ Lq(0, T ; X∗), where 1p +1q = 1.
(vii) If X is a reflexive, separable Banach space, then for any bounded sequence {vn} in L∞(0, T ; X), there exists v ∈ L∞(0, T ; X) and a subsequence {vnk} ⊂ {vn} weakly-∗ convergent in L∞(0, T ; X) to v, i.e.
Z T
0 hvnk(t), w(t)iX∗×X dt→ Z T
0 hv(t), w(t)iX∗×X dt for all w∈ L1(0, T ; X∗).
(viii) If 0≤ s ≤ t ≤ T and v ∈ L1(0, T ; X), then
Z t
s
v(τ ) dτ
X
≤ Z t
s kv(τ)kXdτ.
Recall now the definition of the Bochner-Sobolev spaces. Let 1 ≤ p ≤ ∞. By W1,p(0, T ; X) we denote the subspace of Lp(0, T ; X) of functions whose first order weak derivative with respect to time belongs to Lp(0, T ; X), i.e.
W1,p(0, T ; X) ={ u ∈ Lp(0, T ; X)| u′ ∈ Lp(0, T ; X)}.
It is well known (cf. e.g. Chapter 3.4 of [24], Chapter 2 in [25]) that this space endowed with a norm kukW1,p(0,T ;X) =kukLp(0,T ;X)+ku′kLp(0,T ;X) becomes a Banach space and the embedding W1,p(0, T ; X)⊂ C(0, T ; X) is continuous. For the definition and properties of the Bochner-Sobolev spaces Wk,p(0, T ; X) for k ≥ 1, we refer to e.g.
Chapter 23 of [99] and Chapter 3.4 of [24].
Next, we recall facts we need for the understanding of the concept of evolution triple. The space of all linear and continuous operators from a normed space X to a normed space Y will be denoted by L(X, Y ).
Proposition 2 Let X and Y be Banach spaces, and let A ∈ L(X, Y ). Then the dual operator A∗: Y∗ → X∗ is also linear and continuous, and we have kAkL(X,Y ) = kA∗kL(Y∗,X∗). Moreover, if the linear operator A : X → Y is compact, then so is the dual operator A∗.
Proposition 3 Let X and Y be Banach spaces with X ⊂ Y such that X is dense in Y , and the embedding i : X → Y is continuous. Then
(i) the embedding Y∗ ⊂ X∗ is continuous and the embedding operator bi: Y∗ → X∗ coincides with the dual operator of i, i.e. bi = i∗;
(ii) if X is, in addition, reflexive, then Y∗ is dense in X∗;
(iii) if the embedding X ⊂ Y is compact, then so is the embedding Y∗ ⊂ X∗; (iv) for all v∈ L2(0, T ; X), we have kvkL2(0,T ;Y ) ≤ kikL(X,Y )kvkL2(0,T ;X).
The following notion of evolution triple, or sometimes called the Gelfand triple (cf.
Chapter 23 of [99], Chapter 3.4 of [24]), is basic in the study of evolution problems.
Definition 4 A triple of spaces (V, H, V∗) is called an evolution triple if the follow- ing properties hold
(a) V is a separable and reflexive Banach space, and H is separable Hilbert space endowed with the scalar product h·, ·i;
(b) the embedding V ⊂ H is continuous, and V is dense in H;
(c) identifying H with its dual H∗ by the Riesz map, we then have H ⊂ V∗ with the equality hh, viV∗×V =hh, vi for h ∈ H ⊂ V∗, v ∈ V .
Since V is reflexive and V is dense in H, the space H∗ is dense in V∗, and hence, H is dense in V∗.
Example 5 Let Ω ⊂ RN be a bounded domain with Lipschitz boundary and let V be a closed subspace of W1,p(Ω; Rd) with 2 ≤ p < ∞ such that W01,p(Ω; Rd) ⊂ V ⊂ W1,p(Ω; Rd). Then (V, H, V∗) with H = L2(Ω; Rd) is an evolution triple with all em- beddings being, in addition, compact.
Finally, we introduce the Bochner-Sobolev space related to the Gelfand triple. Let (V, H, V∗) be an evolution triple, 1 < p <∞ and 1p + 1q = 1. We set
W1,p(0, T ; V, H) ={ u ∈ Lp(0, T ; V )| u′ ∈ Lq(0, T ; V∗)},
where the time derivative involved in the definition is understood in the sense of vector valued distributions. We equip this space with the following norm
kukW1,p(0,T ;V,H)=kukLp(0,T ;V )+ku′kLq(0,T ;V∗).
It is well known (cf. Proposition 23.23 of [99], Theorem 3.4.13 and Proposition 3.4.14 of [24]) that the embedding W1,p(0, T ; V, H) ⊂ C(0, T ; H) is continuous (precisely, for each u ∈ W1,p(0, T ; V, H) there exists a uniquely determined continuous func- tion u1: [0, T ] → H such that u(t) = u1(t) a.e. t ∈ [0, T ]) and the embedding W1,p(0, T ; V, H)⊂ Lp(0, T ; H) is compact.
In the subsequent sections we will use the following notation for an evolution triple (V, H, V∗) and p = q = 2:
V = L2(0, T ; V ), H = Lb 2(0, T ; H), V∗ = L2(0, T ; V∗),
W = W1,2(0, T ; V, H) ={ v ∈ V | v′ ∈ V∗}.
With the norm introduced above, the space W becomes a separable reflexive Banach space and the following embeddings W ⊂ V ⊂ bH ⊂ V∗, W ⊂ C(0, T ; H) and {w ∈ V | w′ ∈ W} ⊂ C(0, T ; V ) are continuous. By Theorem 5.1 in Chapter 1 of Lions [52] the embedding W ⊂ bH is also compact. The continuity of the embedding W ⊂ C(0, T ; H) entails the following result (cf. Lemma 4(b) of [69]) which will be useful in our study.
Corollary 6 If un, u∈ W and un → u weakly in W, then un(t)→ u(t) weakly in H for all t∈ [0, T ].
Furthermore, given a Banach space Y , we will use the following notation Pf (c)(Y ) = { A ⊆ Y | A is nonempty, closed, (convex) };
P(w)k(c)(Y ) = { A ⊆ Y | A is nonempty, (weakly) compact, (convex) }.
2.2 Single-valued and multivalued operators
Let X be a reflexive Banach space with the norm k · k, X∗ be its dual and let h·, ·i denote the duality pairing of X∗ and X. First we recall some definitions related to the single-valued and multivalued operators (cf. Denkowski et al. [23, 24], Hu and Papa- georgiou [37], Naniewicz and Panagiotopoulos [73], Showalter [94] and Zeidler [99]).
Definition 7 A mapping T from X to X∗ is said to be
(i) bounded if it takes bounded sets of X into bounded sets of X∗;
(ii) weakly (strongly) continuous if for every xn→ x weakly (strongly) in X, we have T xn→ T x weakly (strongly) in X∗;
(iii) hemicontinuous if the real-valued function t→ hT (u + tv), wi is continuous on [0, 1] for all u, v, w∈ X;
(iv) demicontinuous if for every xn→ x in X, we have T xn → T x weakly in X∗; (v) monotone if hT x − T y, x − yi ≥ 0 for all x, y ∈ X;
(vi) maximal monotone if T is monotone and for any x, y ∈ X, w ∈ X∗ such that hT x − w, x − yi ≥ 0, we have w = T y;
(vii) strongly monotone if there exists c > 0 and p > 1 such that for any x, y ∈ X, we have hT x − T y, x − yi ≥ c kx − ykp;
(viii) pseudomonotone if xn→ x weakly in X and lim sup hT xn, xn−xi ≤ 0 implies hT x, x − vi ≤ lim inf hT xn, xn− vi for all v ∈ X.
Remark 8 It can be shown (cf. [11]) that a mapping T : X → X∗ is pseudomono- tone according to (viii) of Definition 7 if and only if xn → x weakly in X and lim suphT xn, xn − xi ≤ 0 implies lim hT xn, xn − xi = 0 and T xn → T x weakly in X∗.
Definition 9 A mapping T from X to 2X∗ is said to be
(i) bounded if set T (C) is bounded in X∗ for any bounded subset C ⊂ X;
(ii) upper semicontinuous if set T−(C) = {x ∈ X | T x ∩ C 6= ∅} is closed in X for any closed subset C ⊂ X∗ (cf. also Definition 77 and Remark 78);
(iii) monotone if for all x, y ∈ X, x∗ ∈ T x, y∗ ∈ T y, we have hx∗− y∗, x− yi ≥ 0;
(iv) maximal monotone if T is monotone and for any x ∈ X, x∗ ∈ X∗ such that hx∗− y∗, x− yi ≥ 0 for all y ∈ Y∗, y∗ ∈ T y, we have x∗ ∈ T x;
(vii) strongly monotone if there exists c > 0 and p > 1 such that for any x, y ∈ X, x∗ ∈ T x, y∗ ∈ T y, we have hx∗ − y∗, x− yi ≥ c kx − ykp;
(v) pseudomonotone if it satisfies
(a) for every x∈ X, T x is a nonempty, convex, and weakly compact set in X∗; (b) T is upper semicontinuous from every finite dimensional subspace of X into
X∗ endowed with the weak topology;
(c) if xn → x weakly in X, x∗n ∈ T xn, and lim suphx∗n, xn−xi ≤ 0, then for each y∈ X there exists x∗(y)∈ T x such that hx∗(y), x− yi ≤ lim inf hx∗n, xn− xi.
(vi) coercive if there exists a function c : R+ → R with lim
r→+∞c(r) = +∞ such that for all x∈ X and x∗ ∈ T x, we have hx∗, xi ≥ c (kxk)kxk;
Let L : D(L)⊂ X → X∗ be a linear densely defined maximal monotone operator. A mapping T : X → 2X∗ is said to be
(vii) L-pseudomonotone (pseudomonotone with respect to D(L)) if and only if (v)(a), (b) and the following hold:
(d) if{xn} ⊂ D(L) is such that xn → x weakly in X, x ∈ D(L), Lxn → Lx weakly in X∗, x∗n ∈ T xn, x∗n → x∗ weakly in X∗, and lim suphx∗n, xn− xi ≤ 0, then x∗ ∈ T x and hx∗n, xni → hx∗, xi.
The following surjectivity result for L-pseudomonotone operators can be found in Theorem 1.3.73 of Denkowski et al. [24] and for the convenience of the reader we include it here.
Theorem 10 If X is a reflexive, strictly convex Banach space, L : D(L)⊂ X → X∗ is a linear densely defined maximal monotone operator, and T : X → 2X∗ \ {∅} is bounded, coercive and pseudomonotone with respect to D(L), then L + T is surjective.
Finally, we recall a result which show that certain properties of the operator A are transferred to its Nemitsky (superposition) operator bA.
Lemma 11 Let V be a reflexive Banach space with the norm k · k, the dual V∗ and let h·, ·i denote the duality pairing of V∗ and V . Let 2 ≤ p < ∞, 1p + 1q = 1 and let A : (0, T )× V → V∗ be an operator such that
(i) A(·, v) is measurable on (0, T), for all v ∈ V ; (ii) A(t,·) is demicontinuous, for a.e. t ∈ (0, T );
(iii) there exists a nonnegative function a1 ∈ Lq(0, T ) and a constant b1 > 0 such that kA(t, v)kV∗ ≤ a1(t) + b1kvkp−1 for all v ∈ V, a.e. t ∈ (0, T );
(iv) there exist constants b2 > 0, b3 ≥ 0 and a function a2 ∈ L1(0, T ) such that hA(t, v), vi ≥ b2kvkp− b3kvkr− a2(t)
for all v∈ V and a.e. t ∈ (0, T ) with p > r.
Then the Nemitsky operator bA : Lp(0, T ; V )→ Lq(0, T ; V∗) defined by ( bAv)(t) = A(t, v(t)) for v ∈ Lp(0, T ; V )
has the following properties:
(i) bA is well defined, i.e. bAv ∈ Lq(0, T ; V∗) for all v∈ Lp(0, T ; V );
(ii) bA is demicontinuous;
(iii) there exist constants ba1 ≥ 0 and bb1 > 0 such that
k bAvkLq(0,T ;V∗) ≤ ba1+ bb1kvkp−1Lp(0,T ;V ) for all v∈ Lp(0, T ; V );
(iv) there exist constants a2 > 0 and b2 ≥ 0 such that
h bAv, viLq(0,T ;V∗)×Lp(0,T ;V ) ≥ b2kvkpLp(0,T ;V )− bb2kvkrLp(0,T ;V )− ba2 for all v∈ Lp(0, T ; V ).
For the proof of the above lemma, we refer to Berkovits and Mustonen [11], and Ochal [75].
2.3 Clarke’s generalized subdifferential
The purpose of this section is to present the basic facts of the theory of generali- zed differentiation for a locally Lipschitz function (cf. Clarke [21], Clarke et al. [22], Denkowski et al. [23] and Hu and Papageorgiou [37]). We also elaborate on the classes of functions which are regular in the sense of Clarke and prove a few results needed in what follows. Throughtout this section X is a Banach space, X∗ is its dual and h·, ·iX∗×X denotes the duality pairing between X∗ and X.
Definition 12 (Locally Lipschitz function) A function ϕ : U → R defined on an open subset U of X is said to be locally Lipschitz on U, if for each x0 ∈ U there exists K > 0 and ε > 0 such that
|ϕ(y) − ϕ(z)| ≤ Kky − zk for all y, z ∈ B(x0, ε).
A function ϕ : U ⊆ X → R, which is Lipschitz continuous on bounded subsets of U is locally Lipschitz. The converse assertion is not generally true, cf. Chapter 2.5 of Carl et al. [16].
Definition 13 (Generalized directional derivative) The generalized directional derivative (in the sense of Clarke) of the locally Lipschitz function ϕ : U → R at the point x∈ U in the direction v ∈ X, denoted by ϕ0(x; v), is defined by
ϕ0(x; v) = lim sup
y→x, λ↓0
ϕ(y + λv)− ϕ(y)
λ .
We observe that in contrast to the usual directional derivative, the generalized directional derivative ϕ0 is always defined.
Definition 14 (Generalized gradient) Let ϕ : U → R be a locally Lipschitz func- tion on an open set U of X. The generalized gradient (in the sense of Clarke) of ϕ at x∈ U, denoted by ∂ϕ(x), is a subset of a dual space X∗ defined as follows
∂ϕ(x) ={ ζ ∈ X∗ | ϕ0(x; v)≥ hζ, viX∗×X for all v ∈ X }.
The next proposition provides basic properties of the generalized directional deriva- tive and the generalized gradient.
Proposition 15 If ϕ : U → R is a locally Lipschitz function on an open set U of X, then
(i) for every x∈ U the function X ∋ v → ϕ0(x; v)∈ R is sublinear, finite, positively homogeneous, subadditive, Lipschitz continuous and ϕ0(x;−v) = (−ϕ)0(x; v) for all v∈ X;
(ii) the function U × X ∋ (x, v) → ϕ0(x; v)∈ R is upper semicontinuous, i.e. for all x ∈ U, v ∈ X, {xn} ⊂ U, {vn} ⊂ X, xn → x in U and vn → v in X, we have lim sup ϕ0(xn; vn)≤ ϕ0(x; v);
(iii) for every v∈ X we have ϕ0(x; v) = max{ hz, vi | z ∈ ∂ϕ(x) };
(iv) for every x ∈ U the gradient ∂ϕ(x) is nonempty, convex, and weakly-∗ compact subset of X∗ which is bounded by the Lipschitz constant K > 0 of ϕ near x;
(v) the graph of the generalized gradient ∂ϕ is closed in U× (w-∗-X∗)-topology, i.e.
if {xn} ⊂ U and {ζn} ⊂ X∗ are sequences such that ζn ∈ ∂ϕ(xn) and xn→ x in X, ζn → ζ weakly-∗ in X∗, then ζ ∈ ∂ϕ(x), where (w-∗-X∗) denotes the space X∗ equipped with weak-∗ topology;
(vi) the multifunction U ∋ x → ∂ϕ(x) ⊆ X∗ is upper semicontinuous from U into w-∗-X∗.
Proof. The properties (i)-(v) can be found in Propositions 2.1.1, 2.1.2 and 2.1.5 of Clarke [21]. For the proof of (vi), we observe that from (iii), the multifunction ∂ϕ is locally relatively compact (i.e. for every x ∈ X, there exists a neighborhood Ux of
x such that ∂ϕ(Ux) is a weakly-∗ compact subset of X∗). Thus, due to Proposition 4.1.16 of [23], since the graph of ∂ϕ is closed in X× (w-∗-X∗)-topology, we obtain the upper semicontinuity of x7→ ∂ϕ(x).
In order to state the relations between the generalized directional derivative and classical notions of differentiability, we need the following.
Definition 16 (Classical (one-sided) directional derivative) Let ϕ : U → R be defined on an open subset U of X. The directional derivative of ϕ at x∈ U in the direction v ∈ X is defined by
ϕ′(x; v) = lim
λ↓0
ϕ(x + λv)− ϕ(x)
λ , (2)
when the limit exists.
We recall the definition of a regular function which is needed in the sequel.
Definition 17 (Regular function) A function ϕ : U → R on an open set U of X is said to be regular (in the sense of Clarke) at x∈ U, if
(i) for all v∈ X the directional derivative ϕ′(x; v) exists, and (ii) for all v ∈ X, ϕ′(x; v) = ϕ0(x; v).
The function ϕ is regular (in the sense of Clarke) on U if it is regular at every point x∈ U.
Remark 18 Directly from Definitions 13 and 16, it is clear that ϕ′(x; v)≤ ϕ0(x; v) for all x∈ U and all v ∈ X when ϕ′(x; v) exists.
Definition 19 (Gˆateaux derivative) Let ϕ : U → R be defined on an open subset U of X. We say that ϕ is Gˆateaux differentiable at x∈ U provided that the limit in (2) exists for all v ∈ X and there exists a (necessarily unique) element ϕ′G(x)∈ X∗ (called the Gˆateaux derivative) that satisfies
ϕ′(x; v) =hϕ′G(x), viX∗×X for all v ∈ X. (3) Definition 20 (Fr´echet derivative) Let ϕ : U → R be defined on an open subset U of X. We say that ϕ is Fr´echet differentiable at x ∈ U provided that (3) holds at the point x and in addition that the convergence in (2) is uniform with respect to v in bounded subsets of X. In this case, we write ϕ′(x) (the Fr´echet derivative) in place of ϕ′G(x).
The two notions of differentiability are not equivalent, even in finite dimensions.
The following relations between Gˆateaux and Fr´echet derivative hold. If ϕ is Fr´echet differentiable at x ∈ U, then ϕ is Gˆateaux differentiable at x. If ϕ is Gˆateaux dif- ferentiable in a neighborhood of x0 and ϕ′G is continuous at x0, then ϕ is Fr´echet differentiable at x0 and ϕ′(x0) = ϕ′G(x0).
Remark 21 If ϕ : U ⊂ X → R is Fr´echet differentiable in U and ϕ′(·): U → X∗ is continuous, then we say that ϕ is continuously differentiable and write ϕ∈ C1(U).
The following notion of strict differentiability is intermediate between Gˆateaux and continuous differentiability. It is known that the Clarke subdifferential ∂ϕ(x) reduces to a singleton precisely when ϕ is strictly differentiable.
Definition 22 (Strict differentiability) A function ϕ : U → R be defined on an open subset U of X is strictly (Hadamard) differentiable at x ∈ U, if there exists an element Dsϕ(x)∈ X∗ such that
y→x, λ↓0lim
ϕ(y + λv)− ϕ(y)
λ =hDsϕ(x), viX∗×X for all v ∈ X and provided the convergence is uniform for v in compact sets.
The following notion of subgradient of convex function generalizes the classical concept of a derivative.
Definition 23 (Convex subdifferential) Let U be a convex subset of X and ϕ : U → R be a convex function. An element x∗ ∈ X∗ is called a subgradient of ϕ at x∈ X if and only if the following inequality holds
ϕ(v)≥ ϕ(x) + hx∗, v− xiX∗×X for all v ∈ X. (4) The set of all x∗ ∈ X∗ satisfying (4) is called the subdifferential of ϕ at x, and is denoted by ∂ϕ(x).
The following two propositions follow from Chapters 2.2 and 2.3 of [21].
Proposition 24 Let ϕ : U → R be defined on an open subset U of X. Then
(i) the function ϕ is strictly differentiable at x∈ U if and only if ϕ is locally Lipschitz near x and ∂ϕ(x) is a singleton (which is necessarily the strict derivative of ϕ at x). In particular, if ϕ is continuously differentiable at x∈ U, then ϕ0(x, v) = ϕ′(x; v) =hϕ′(x), viX∗×X for all v ∈ X and ∂ϕ(x) = {ϕ′(x)};
(ii) if ϕ is regular at x∈ U and ϕ′(x) exists, then ϕ is strictly differentiable at x;
(iii) if ϕ is regular at x ∈ U, ϕ′(x) exists and g is locally Lipschitz near x, then
∂(ϕ + g)(x) ={ϕ′(x)} + ∂g(x);
(iv) if ϕ is Gˆateaux differentiable at x∈ U, then ϕ′G(x)∈ ∂ϕ(x);
(v) if U is a convex set and ϕ : U → R is convex, then the Clarke subdifferential
∂ϕ(x) at any x ∈ U coincides with the subdifferential of ϕ at x in the sense of convex analysis.
(vi) if U is a convex set and ϕ : U → R is convex, then the Clarke subdifferential
∂ϕ : U → 2X∗ is a monotone operator.
The following result collects the properties of regular functions.
Proposition 25
(i) If ϕ : U → R defined on an open subset U of X is strictly differentiable at x ∈ U, then ϕ is regular at x;
(ii) If the open set U is convex and ϕ : U → R is a convex function, then ϕ is locally Lipschitz and regular on U;
(iii) Any finite nonnegative linear combination of regular functions at x, is regular at x;
(iv) If ϕ : U → R defined on an open subset U of X is regular at x ∈ U and there exists the Gˆateaux derivative ϕ′G(x) of ϕ at x, then ∂ϕ(x) ={ϕ′G(x)}.
In the case X is of finite dimension, we have the following characterization of the Clarke subdifferential (cf. Theorem 2.5.1 of [21]). Recall that if a function ϕ : Rn→ R is Lipschitz on an open set U ⊂ Rn, then by the celebrated theorem of Rademacher (cf. e.g. Corollary 4.19 in [22]), ϕ is Fr´echet differentiable almost everywhere on U.
Proposition 26 Let ϕ : U ⊂ Rn → R be a locally Lipschitz near x ∈ U, N be any Lebesgue-null set in Rn and Nϕ be the Lebesgue-null set outside of which ϕ is Fr´echet differentiable. Then
∂ϕ(x) = co{ lim ∇ϕ(xi)| xi → x, xi ∈ N, x/ i ∈ N/ ϕ}.
Now we recall the basic calculus rules for the generalized directional derivative and the generalized gradient which are needed in the sequel.
Proposition 27 (i) For a locally Lipschitz function ϕ : U → R defined on an open subset U of X and for all λ∈ R, we have ∂(λϕ)(x) = λ∂ϕ(x) for all x ∈ U;
(ii) (The sum rules) For locally Lipschitz functions ϕ1, ϕ2: U → R defined on an open subset U of X, we have
∂(ϕ1+ ϕ2)(x) ⊂ ∂ϕ1(x) + ∂ϕ2(x) for all x∈ U (5) or equivalently
(ϕ1+ ϕ2)0(x; v)≤ ϕ01(x; v) + ϕ02(x; v) for all v ∈ X; (6) (iii) If one of ϕ1, ϕ2 is strictly differentiable at x∈ U, then in (5) and (6) equalities
hold.
(iv) In addition, if ϕ1, ϕ2 are regular at x∈ U, then ϕ1+ ϕ2 is regular and we also have equalities in (5) and (6). The extension of (5) and (6) to finite nonnegative linear combinations is immediate.
Proposition 28 Let X and Y be Banach spaces, A ∈ L(Y, X) and let ϕ: X → R be a locally Lipschitz function. Then
(a) (ϕ◦ A)0(x; v)≤ ϕ0(Ax; Av) for all x, v∈ Y ,
(b) ∂(ϕ◦ A)(x) ⊆ A∗∂ϕ(Ax) for all x∈ Y,
where A∗ ∈ L(X∗, Y∗) denotes the adjoint operator to A. If in addition either ϕ or
−ϕ is regular at Ax, then either ϕ ◦ A: Y → R or (−ϕ) ◦ A: Y → R is regular and (a) and (b) hold with equalities. The equalities in (a) and (b) are also true if, instead of the regularity condition, we assume that A is surjective.
Proposition 29 Let X1 and X2 be Banach spaces. If ϕ : X1 × X2 → R is locally Lipschitz and regular at x = (x1, x2)∈ X1 × X2, then
∂ϕ(x1, x2)⊂ ∂1ϕ(x1, x2)× ∂2ϕ(x1, x2), (7) where by ∂1ϕ(x1, x2) (respectively ∂2ϕ(x1, x2)) we denote the partial generalized sub- differential of ϕ(·, x2) (respectively ϕ(x1,·)), or equivalently
ϕ0(x1, x2; v1, v2)≤ ϕ01(x1, x2; v1) + ϕ02(x1, x2; v2) for all (v1, v2)∈ X1× X2, where ϕ01(x1, x2; v1) (respectively ϕ02(x1, x2; v2)) denotes the partial generalized direc- tional derivative of ϕ(·, x2) (respectively ϕ(x1,·)) at the point x1 (respectively x2) in the direction v1 (respectively v2).
In general in Proposition 29, without the regularity hypothesis, there is no relation between the two sets in (7), cf. Example 2.5.2 in [21].
Lemma 30 Let X1 and X2 be Banach spaces and let ϕ : X1 × X2 → R be locally Lipschitz function at (x1, x2)∈ X1 × X2.
(1) If g : X1 → R is locally Lipschitz at x1 and ϕ(y1, y2) = g(y1) for all (y1, y2)∈ X1× X2, then
(i) ϕ0(x1, x2; v1, v2) = g0(x1; v1) for all (v1, v2)∈ X1× X2; (ii) ∂ϕ(x1, x2) = ∂g(x1)× {0}.
(2) If h : X2 → R is locally Lipschitz at x2 and ϕ(y1, y2) = h(y2) for all (y1, y2)∈ X1× X2, then
(i) ϕ0(x1, x2; v1, v2) = h0(x2; v2) for all (v1, v2)∈ X1× X2; (ii) ∂ϕ(x1, x2) = {0} × ∂h(x2).
Proof. We prove (1) since the proof of (2) is analogous. The first relation follows from the direct calculation
ϕ0(x1, x2; v1, v2) = lim sup
(y1,y2)→(x1,x2), λ↓0
ϕ((y1, y2) + λ(v1, v2))− ϕ(y1, y2)
λ =
= lim sup
(y1,y2)→(x1,x2), λ↓0
g(y1+ λv1)− g(y1)
λ =
= lim sup
y1→x1, λ↓0
g(y1+ λv1)− g(y1)
λ = g0(x1; v1)
for all (v1, v2) ∈ X1 × X2. For the proof of (ii), let (x∗1, x∗2) ∈ ∂ϕ(x1, x2). By the definition, we have
hx∗1, v1iX1∗×X1 +hx∗2, v2iX2∗×X2 ≤ ϕ0(x1, x2; v1, v2)
for every (v1, v2) ∈ X1 × X2. Choosing (v1, v2) = (v1, 0), we obtain hx∗1, v1iX1∗×X1 ≤ ϕ0(x1, x2; v1, 0) = g0(x1; v1) for every v1 ∈ X1 which means that x∗1 ∈ ∂g(x1). Taking (v1, v2) = (0, v2), we get hx∗2, v2iX2∗×X2 ≤ g0(x1; 0) = 0 for v2 ∈ X2. Since v2 ∈ X2 is arbitrary, we have hx∗2, v2iX2∗×X2 = 0 and then x∗2 = 0.
Conversely, let (x∗1, x∗2)∈ ∂g(x1)× {0}. For all (v1, v2)∈ X1× X2, we have hx∗1, v1iX1∗×X1 +hx∗2, v2iX2∗×X2 =hx∗1, v1iX1∗×X1 ≤ g0(x1; v1) = ϕ0(x1, x2; v1, v2) which implies that (x∗1, x∗2)∈ ∂ϕ(x1, x2). The proof is complete.
Next, we elaborate on locally Lipschitz functions which are regular in the sense of Clarke. We consider the classes of max (min) type and d.c type (difference of convex functions). The proof of the first result can be found in Proposition 2.3.12 of [21] and Proposition 5.6.29 of [23].
Proposition 31 Let ϕ1, ϕ2: U → R be locally Lipschitz functions near x ∈ U, U be an open subset of X and ϕ = max{ϕ1, ϕ2}. Then ϕ is locally Lipschitz near x and
∂ϕ(x)⊂ co {∂ϕk(x)| k ∈ I(x)}, (8)
where I(x) = {k ∈ {1, 2} | ϕ(x) = ϕk(x)} is the active index set at x. If in addition, ϕ1 and ϕ2 are regular at x, then ϕ is regular at x and (8) holds with equality.
Corollary 32 Let ϕ1, ϕ2: U → R be strictly differentiable functions at x ∈ U, U be an open subset of X and ϕ = min{ϕ1, ϕ2}. Then −ϕ is locally Lipschitz near x, regular at x and ∂ϕ(x) = co{∂ϕk(x) | k ∈ I(x)}, where I(x) is the active index set at x.
Proof. Since ϕ1 and ϕ2 are strictly differentiable at x∈ U, the functions −ϕ1 and
−ϕ2 also have the same property. From Proposition 25(i), it follows that −ϕ1 and
−ϕ2 are locally Lipschitz near x and regular at x. Let g1 = −ϕ1, g2 = −ϕ2 and g = max{g1, g2}. It follows from Proposition 31 that g is locally Lipschitz near x, regular at x and ∂g(x) = co{∂gk(x)| k ∈ I(x)}. On the other hand, we have
g = max{g1, g2} = max{−ϕ1,−ϕ2} = − min{ϕ1, ϕ2} = −ϕ and
−∂ϕ(x) = ∂(−ϕ)(x) = ∂g(x) = co {∂(−ϕk)(x) | k ∈ I(x)} =
= co{−∂ϕk(x)| k ∈ I(x)} = −co {∂ϕk(x)| k ∈ I(x)}.
Hence the conclusion of the corollary follows.
The next proposition generalizes Lemma 14 of [68].
Proposition 33 Let ϕ1, ϕ2: U → R be convex functions, U be an open convex subset of X, ϕ = ϕ1− ϕ2 and x∈ U. Assume that
∂ϕ1(x) is singleton (or ∂ϕ2(x) is singleton).
Then
−ϕ is regular at x (or ϕ is regular at x respectively) and
∂ϕ(x) = ∂ϕ1(x)− ∂ϕ2(x), (9)
where ∂ϕk, k = 1, 2 are the subdifferentials in the sense of convex analysis.
Proof. From Proposition 25(ii) we know that ϕk, k = 1, 2 are locally Lipschitz and regular on U. Suppose ∂ϕ1(x) is a singleton. By Proposition 24(i), the function ϕ1
is strictly differentiable at x. Thus −ϕ1 is also strictly differentiable at x and again, by Proposition 25(ii), it follows that −ϕ1 is regular at x. Hence −ϕ = −ϕ1 + ϕ2 is regular at x as the sum of two regular functions. Moreover, from Propositions 25(iii) and 27, we have
−∂ϕ(x) = ∂(−ϕ)(x) = ∂(−ϕ1+ ϕ2)(x) =
= ∂(−ϕ1)(x) + ∂ϕ2(x) =−∂ϕ1(x) + ∂ϕ2(x) which implies (9).
If ∂ϕ2(x) is a singleton, then as before by using Proposition 24(i), (ii), we deduce ϕ2 is strictly differentiable at x which in turn implies that−ϕ2 is strictly differentiable and regular at x. So ϕ = ϕ1 + (−ϕ2) is regular at x being the sum of two regular functions and by Propositions 25(iii) and 27, we obtain
∂ϕ(x) = ∂(ϕ1+ (−ϕ2))(x) = ∂ϕ1(x) + ∂(−ϕ2)(x) = ∂ϕ1(x)− ∂ϕ2(x)
which gives the equality (9). In view of convexity of ϕk, k = 1, 2 their Clarke subdif- ferentials coincide with the subdifferentials in the sense of convex analysis. The proof is completed.
Lemma 34 Let X and Y be Banach spaces and ϕ : X × Y → R be such that (i) ϕ(·, y) is continuous for all y ∈ Y ;
(ii) ϕ(x,·) is locally Lipschitz on Y for all x ∈ X;
(iii) there is a constant c > 0 such that for all η ∈ ∂ϕ(x, y), we have kηkY∗ ≤ c (1 + kxkX +kykY) for all x∈ X, y ∈ Y, where ∂ϕ denotes the generalized gradient of ϕ(x,·).
Then ϕ is continuous on X × Y .
Proof. Let x ∈ X and y1, y2 ∈ Y . By the Lebourg mean value theorem (cf. e.g.
Theorem 5.6.25 of [23]), we can find y∗ in the interval [y1, y2] and u∗ ∈ ∂ϕ(x, y∗) such that ϕ(x, y1)− ϕ(x, y2) =hu∗, y1− y2iY∗×Y. Hence
|ϕ(x, y1)− ϕ(x, y2)| ≤ ku∗kY∗ky1− y2kY ≤
≤ c (1 + kxkX +ky∗kY)ky1− y2kY ≤
≤ c1(1 +kxkX +ky1kY +ky2kY)ky1− y2kY
for some c1 > 0. Let{xn} ⊂ X and {yn} ⊂ Y be such that xn→ x0 in X and yn→ y0
in Y . We have
|ϕ(xn, yn)− ϕ(x0, y0)| ≤ |ϕ(xn, yn)− ϕ(xn, y0)| + |ϕ(xn, y0)− ϕ(x0, y0)| ≤
≤ c1(1 +kxnkX +kynkY +ky0kY)kyn− y0kY + + |ϕ(xn, y0)− ϕ(x0, y0)|.
SincekxnkX,kynkY ≤ c2 with a constant c2 > 0 and ϕ(·, y0) is continuous, we deduce that ϕ(xn, yn)→ ϕ(x0, y0), which completes the proof.
We conclude this section with a result on measurability of the multifunction of the subdifferential type.
Proposition 35 Let X be a separable reflexive Banach space, 0 < T < ∞ and ϕ : (0, T )× X → R be a function such that ϕ(·, x) is measurable for all x ∈ X and ϕ(t,·) is locally Lipschitz for all t ∈ (0, T ). Then the multifunction (0, T ) × X ∋ (t, x) 7→ ∂ϕ(t, x) ⊂ X∗ is measurable, where ∂ϕ denotes the Clarke generalized gradient of ϕ(t,·).
Proof. Let (t, x) ∈ (0, T ) × X. First note that by Definition 13, we may express the generalized directional derivative of ϕ(t,·) as the upper limit of the quotient
1
λ(ϕ(t, y + λv)− ϕ(t, y)), y ∈ X, where λ ↓ 0 taking rational values and y → x taking values in a countable dense subset of X (recall that X is separable):
ϕ0(t, x; v) = lim sup
y→x, λ↓0
ϕ(t, y + λv)− ϕ(t, y)
λ = inf
r>0 sup
ky − xk ≤ r 0 < λ < r
ϕ(t, y + λv)− ϕ(t, y) λ
= inf
r>0 sup
ky − xk ≤ r, 0 < λ < r y∈ D, λ ∈ Q
ϕ(t, y + λv)− ϕ(t, y) λ
for all v ∈ X, where D ⊂ X is a countable dense set. From this it follows that the function (t, x, v) 7→ ϕ0(t, x; v) is Borel measurable as ”the countable” limsup of measurable functions of (t, x, v) (note that by hypotheses, the function (t, x) 7→
ϕ(t, x) being Carath´eodory, it is jointly measurable). From Lemma 69, it follows that (t, x)7→ ϕ0(t, x; v) is measurable for every v ∈ X.
